A technique known as filtered back-projection (FBP) is nowadays used as the standard method for reconstructing computed tomography image data from a CT scanner's X-ray CT data sets for which, during data acquisition, an X-ray source from which conical or more specifically pyramidal X-ray beams are emitted rotates around a VoI on a helical path. With this method, the data is first pre-processed in order to make it as noise-free as possible. What is termed a “rebinning” step is then carried out in which the data generated with the beam fanning out from the source is rearranged such that it is present in a form as if the detector had been hit by an X-ray wave front approaching the detector in a parallel manner. The data is then transformed in the frequency domain. Filtering with what is termed a ramp filter takes place in the frequency domain and the filtered data is then inverse-transformed. Using the thus re-sorted and filtered data, back-projection onto the individual voxels within the volume of interest then takes place. Although this method works very well in principle, no mathematically exact reconstructions are possible, which may result in artifacts. In particular, because of their approximative mode of operation, the conventional FBP methods are subject to problems with low-frequency so-called cone-beam artifacts, particularly if the number of detector rows increases considerably and exceeds 100, as may be the case with the latest detectors.
A method has therefore been developed with which mathematically exact and stable reconstruction is possible even if there are incomplete projections among the incoming projection data. With this method, “differential back-projection” along so-called “π-lines” is performed. In this context, π-lines denote straight lines which intersect the helical path twice within one complete revolution. The back-projection data thereby obtained corresponds to the Hilbert transform of the desired image data, so that the desired image data can be generated by subsequent inverse Hilbert transformation. This method of three-dimensional differential back-projection followed by inverse Hilbert transformation (DBP-HT) is described in more detail in the publication by H. Schöndube, K. Stiersdorfer, F. Dennerlein, T. White and F. Noo: “Towards an efficient two-step hilbert algorithm for helical cone-beam CT.” in Proc. 2007 Meeting on Fully 3D Image Reconstruction in Radiology and Nuclear Medicine (Lindau, Germany), F. Beckman and M. Kachelrieβ, Eds., 2007, pp. 120-123. As the method is theoretically mathematically exact and no artifacts due to the fan beam geometry of the X-ray beam can occur, it can be used to reconstruct good images even when the number of detector rows is markedly increased, as described above.
However, these reconstruction methods have so far been limited to using detector measurement data which is present on the detector within the so-called Tam-Danielsson window (also referred to hereinafter as the “TD window” for short), said TD window being defined by the projection of the X-ray source trajectory, i.e. the helical orbit of the X-ray source, onto the detector. Data that is measured in detector areas outside said TD window has hitherto been unusable in the DBP-HT method. However, since in normal CT systems the conical beam of the X-ray source is implemented such that it is incident on the entire detector, dose is consequently wasted. Although it would be theoretically possible to implement an X-ray source such that it produces an X-ray beam that is precisely incident on the TD window, this would mean no more flexibility in setting the pitch of the helical curve and would also be very cost-intensive, so that it makes more sense to use a rectangular standard detector and a conventional X-ray source. Apart from that, it would be very desirable to be able to use data outside the TD window. This data is redundant data which does not necessarily have to be used for a complete reconstruction. On the other hand, however, the redundant data is very useful for reducing image noise without affecting the resolution of the images. The redundant data could in principle be used with approximative methods, but again at the expense of the advantage of the mathematically exact reconstruction described above. Although an approach for exact mathematical reconstruction in which detector data outside the TD window could also be used is already described in the publication by J. Pack, F. Noo and R. Clackdoyle: “Cone-beam reconstruction using the backprojection of locally filtered projections” IEEE Trans. Med. Imag., vol. 24, no. 1, pp. 70-85, January 2005, this method is limited to performing differential back-projection and subsequent inverse Hilbert transformation for each individual voxel, multiple reconstructions and then averaging of the reconstruction values of the individual voxels being necessary in each case for the individual voxels. This results in a considerable computational complexity in order to reconstruct a complete volume of interest. This method is therefore very inefficient and hardly usable in day-to-day practice.